NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  necon3abid GIF version

Theorem necon3abid 2549
Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
Hypothesis
Ref Expression
necon3abid.1 (φ → (A = Bψ))
Assertion
Ref Expression
necon3abid (φ → (AB ↔ ¬ ψ))

Proof of Theorem necon3abid
StepHypRef Expression
1 df-ne 2518 . 2 (AB ↔ ¬ A = B)
2 necon3abid.1 . . 3 (φ → (A = Bψ))
32notbid 285 . 2 (φ → (¬ A = B ↔ ¬ ψ))
41, 3syl5bb 248 1 (φ → (AB ↔ ¬ ψ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   = wceq 1642  wne 2516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 177  df-ne 2518
This theorem is referenced by:  necon3bbid  2550  ncpw1pwneg  6201  ltlenlec  6207
  Copyright terms: Public domain W3C validator